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Persistence, extinction, and critical patch number for island populations. (English) Zbl 0603.92019

Sufficient conditions are derived for persistence and extinction of a population inhabiting several islands. Discrete reaction-diffusion population models are analyzed which describe growth and diffusion of a population on a group of islands or a patch environment.

A critical patch number is defined as the number of islands below which the population goes extinct on that group of islands. It is shown that population persistence on one island leads to population persistence for the entire archipelago. Both single-species and multi-species models are discussed.

MSC:
92D40Ecology
39A12Discrete version of topics in analysis
References:
[1]Allen, L. J. S.: Persistence and extinction in Lotka-Volterra reaction-diffusion equations. Math. Biosci. 65, 1–12 (1983) · Zbl 0522.92021 · doi:10.1016/0025-5564(83)90068-8
[2]Allen, L. J. S.: Persistence and extinction in single-species reaction-diffusion models. Bull. Math. Biol. 45, 209–227 (1983)
[3]DeAngelis, D. L., Post, W. M., Travis, C. C.: Positive feedback in natural systems. Biomathematics 15. Berlin, Heidelberg, New York: Springer 1986
[4]Fahrig, L., Merriam, G.: Habitat patch connectivity and population survival. Ecology 66, 1762- 1768 (1985) · doi:10.2307/2937372
[5]Gurney, W. S. C., Nisbet, R. M.: The regulation of inhomogeneous populations. J. Theor. Biol. 52, 441–457 (1975) · doi:10.1016/0022-5193(75)90011-9
[6]Hastings, A.: Global stability in Lotka-Volterra systems with diffusion. J. Math. Biol. 6, 163–168 (1978) · Zbl 0393.92013 · doi:10.1007/BF02450786
[7]Hastings, A.: Dynamics of a single species in a spatially varying environment: The stabilizing role of high dispersal rates. J. Math. Biol. 16, 49–55 (1982) · Zbl 0496.92010 · doi:10.1007/BF00275160
[8]Kierstead, H., Slobodkin, L. B.: The size of water masses containing plankton blooms. J. Mar. Res. 12, 141–147 (1953)
[9]Lakshmikantham, V., Leela, S.: Differential and integral inequalities theory and applications, vol. 1, New York: Academic Press 1969
[10]Lancaster, P.: Theory of matrices. New York: Academic Press 1969
[11]Levin, S. A.: Dispersion and population interactions. Am. Nat. 108, 207–228 (1974) · doi:10.1086/282900
[12]Levin, S. A.: Population models and community structure in heterogeneous environments. In: Levin, S. A. (ed.) Studies in mathematical biology: Populations and Communities, vol. II, pp. 439–476. Washington D.C.: M.A.A. 1978
[13]Ludwig, D., Jones, D. D., Holling, C. S.: Qualitative analysis of insect outbreak systems, the spruce budworm and forest. J. Anim. Ecol. 47, 315–332 (1978) · doi:10.2307/3939
[14]Ludwig, D., Aronson, D. G., Weinberger, H. F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217–258 (1979) · Zbl 0412.92020 · doi:10.1007/BF00276310
[15]MacArthur, R. H., Wilson, E. O.: An equilibrium theory of insular zoogeography. Evol. 17, 373–387 (1963) · doi:10.2307/2407089
[16]MacArthur, R. H., Wilson, E. O.: The theory of island biogeography. Princeton, N.J.: Princeton University Press 1967
[17]May, R. M.: Stability and complexity in model ecosystems. Princeton, N.J.: Princeton University Press 1974.
[18]Namba, T.: Asymptotic behaviour of solutions of the diffusive Lotka-Volterra equations. J. Math. Biol. 10, 295–303 (1980) · Zbl 0508.34039 · doi:10.1007/BF00276988
[19]Okubo, A.: Critical patch size for plankton and patchiness. In: Levin, S. A. (ed.) Mathematical Ecology. Proceedings, Trieste 1982 (Lect. Notes Biomath., vol. 54, pp. 456–477) Berlin, Heidelberg, New York: Springer 1984
[20]Othmer, H. G., Scriven, L. E.: Instability and dynamic patterns in cellular networks. J. Theor. Biol. 32, 507–537 (1971) · doi:10.1016/0022-5193(71)90154-8
[21]Skellam, J. G.: Random dispersal in theoretical populations. Biometrika. 38, 196–218 (1951)
[22]Svirezhev, Y. M., Logofet, D. O.: Stability of biological communities. Moscow: Mir Publishers 1983
[23]Yodzis, P.: Competition for space and the structure of ecological communities. Lect. Notes Biomath. 25. Berlin, Heidelberg, New York: Springer 1978